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DiPerna-Majda measures and uniform integrability. (English) Zbl 0970.49012

The topic of uniform integrability is discussed from the point of view of relations to Young measures and DiPerna-Majda measures. A formula for the Rosenthal modulus of uniform integrability of \(\{|u_k|^p\}\) is obtained, namely \[ \eta (\{|u_k|^p\})=\sup _{(\sigma ,\widehat \nu)\in \mathcal U}\int _{\bar \Omega } \int _{\beta _{\mathcal R}{\mathbb R}^m\setminus {\mathbb R}^m}\widehat \nu (d\lambda)\sigma (dx), \] where \(\mathcal U\) is the set of all DiPerna-Majda measures \((\sigma ,\widehat \eta)\) that are generated by the same subsequence of \(\{u_k\}\) and computed with respect to a given ring \(\mathcal R\) of bounded continuous functions on \({\mathbb R}^m\). As an application, the following inequality related to Fatou’s lemma is established: \[ \int _{\Omega }\liminf _{k\to \infty }|u_k(x)|^p dx \leq \inf _{\nu \in \mathcal V}\int _{\Omega }\int {{\mathbb R}^m}|\lambda |^p\nu _x(d\lambda) dx \leq \liminf _{k\to \infty }\int _{\Omega }|u_k(x)|^p dx , \] where \(\mathcal V\) is the set of all Young measures generated by some subsequence of \(\{u_k\}\).
Reviewer: Jan Malý (Praha)

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
49Q20 Variational problems in a geometric measure-theoretic setting
28A20 Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence
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